I am teaching linear algebra our of the book by Fraleigh and Beauregard. We are on “subspaces” (subsets of for now) and a subspace is defined to be a set of vectors that is closed under both vector addition and scalar multiplication. Here are a couple of examples of non-subspaces:
1. . Now this space IS closed under scalar multiplication, note that this space IS closed under additive inverses. But it is not closed under addition as for .
2. (this example is in the book): the vectors are closed under vector addition but not under scalar multiplication.